Evaluate $\displaystyle \lim_{n \to \infty} \sum_{k=1}^n \frac{3k}{2^k}$

Question

Not exactly sure how to approach. Cannot do telescoping, or break up the sum into two parts either, cannot find a way to express division as the sum here. Although, if it was something like $\frac{1}{n(n+1)}$ we could write $\frac{1}{n}-\frac{1}{n+1}$. Would like to solve this with simple/clever arithmetic.

Answer 1

hint

Consider the series of functions

$$\sum \frac{x^k}{2^k}$$

and its derivative.